On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

This paper establishes that sufficiently large topologically generating sets in connected compact, amenable, and reductive algebraic groups are necessarily redundant, providing quantitative bounds linked to finite simple groups of Lie type and demonstrating that these findings partially resolve Gelander's conjectures by showing they follow from the Wiegold conjecture.

The Gist

Imagine you are trying to build a complex machine (a Lie Group) using a collection of tools (a Generating Set).

In the world of mathematics, a "generating set" is a group of elements that, when you combine them in various ways, can create every single part of the machine. The paper asks a very specific question: What is the maximum number of tools you can have in your toolbox such that every single one is absolutely necessary?

If you have a toolbox where you can throw away one tool and still build the whole machine, that toolbox is redundant. If you cannot throw away any tool without breaking the machine, it is irredundant.

The authors, Tal Cohen and Itamar Vigdorovich, are trying to find the "ceiling" on how big this perfect, non-redundant toolbox can be for different types of machines.

Here is the breakdown of their findings using simple analogies:

1. The Two Types of Machines

The paper looks at two main types of mathematical "machines":

• Compact Lie Groups: Think of these as closed, finite loops. They are like a perfect sphere or a donut. You can't go off to infinity; everything stays within a bounded space.
• Amenable Groups: Think of these as flexible, stretchy shapes that don't have "hard" chaotic cores. They are easier to control.

The Big Discovery:
The authors prove that for these "nice" machines (compact and amenable), there is a hard limit on how many tools you can have before you start having extras.

• The Analogy: Imagine trying to fill a backpack. If the backpack is a specific size (determined by the "rank" or complexity of the machine), you can only fit a certain number of items before you are forced to carry duplicates. You can't have an infinite number of unique, necessary items.
• The Contrast: If you have a machine that is not compact (like an infinite straight line), you can keep adding unique, necessary tools forever. The paper confirms that for these "wild" machines, the toolbox can be infinitely large.

2. The "Finite Group" Connection (The Secret Shortcut)

This is the most clever part of the paper. The authors realized that to solve the problem for these complex, continuous machines, they didn't need to look at the machines themselves. Instead, they looked at finite simple groups.

• The Analogy: Imagine you want to know the maximum number of unique ingredients needed to bake a giant, infinite cake. Instead of baking the infinite cake, the authors realized that if you look at the "crumbs" left behind when you bake the cake in different, tiny, finite ovens (finite fields), the rules for the crumbs tell you the rules for the giant cake.
• The Result: They showed that the maximum size of the toolbox for a complex machine is controlled by the maximum size of the toolbox for these tiny, finite "crumb" machines. Since mathematicians already have good estimates for the tiny machines, they can now estimate the big ones.

3. The "Nielsen" Twist (Rearranging the Tools)

The paper also looks at a more sophisticated version of the problem. Imagine your tools aren't just sitting in a pile; you are allowed to mix them, swap them, or combine two tools to make a new one (this is called a Nielsen transformation).

• The Question: Even if you are allowed to shuffle and remix your tools, is there still a limit to how many you can start with before you are guaranteed to have a redundant one?
• The Conjecture: A mathematician named Gelander guessed that for compact machines, the answer is surprisingly small: 2. He thought that if you have 3 or more tools, you can always shuffle them around to find a way to throw one away.
• The Paper's Verdict: They proved this is true for many specific machines (like $SO(3)$ and $SL_2$). They also showed that if a famous unsolved puzzle about finite groups (the Wiegold Conjecture) is true, then Gelander's guess is true for all these machines.

4. Why Does This Matter?

You might ask, "Who cares about the size of a toolbox for abstract math machines?"

• Efficiency: In computer science and cryptography, we often deal with these groups. Knowing the limit on "irredundant" sets helps us design more efficient algorithms. If we know we don't need 100 keys to unlock a door, but only 3, we save time and energy.
• Understanding Structure: It helps mathematicians understand the fundamental "shape" of symmetry. It tells us that while these groups can be incredibly complex, they have a hidden simplicity: they can't be "over-determined" by too many independent parts.

Summary in One Sentence

The paper proves that for "well-behaved" mathematical machines, there is a strict limit on how many unique, essential parts you can have, and this limit is surprisingly low and directly connected to the rules governing tiny, finite versions of those machines.

Technical Summary

Here is a detailed technical summary of the paper "On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups" by Tal Cohen and Itamar Vigdorovich.

1. Problem Statement and Context

The paper investigates the maximal size of irredundant generating sets in topological and algebraic groups.

• Definitions:
  • A subset $X \subset G$ is generating if the only closed subgroup (in the topological case) or Zariski-closed subgroup (in the algebraic case) containing $X$ is $G$ itself.
  • $X$ is irredundant if no proper subset of $X$ generates $G$.
  • $m(G)$ denotes the supremum of the sizes of finite irredundant generating sets.
  • Nielsen-irredundancy: A generating tuple $X$ is Nielsen-irredundant if no automorphism of the free group $F_{|X|}$ applied to $X$ yields a redundant set. The maximal size of such sets is denoted $\mu(G)$.
• The Core Question: For infinite groups (specifically Lie and algebraic groups), is $m(G)$ finite? If so, what are the quantitative bounds?
• Motivation: While $m(G)$ is well-studied for finite groups, infinite groups present challenges. For instance, $m(\mathbb{Z}) = \infty$, and it is known that $m(SL_2(\mathbb{R})) = \infty$. The authors aim to characterize exactly when $m(G)$ is finite and provide explicit bounds, particularly for compact Lie groups and reductive algebraic groups.

2. Key Conjectures and Theoretical Framework

The paper addresses Gelander's Conjecture, which posits that for connected compact simple Lie groups (and simple complex algebraic groups), the Nielsen redundancy rank $\mu(G)$ is exactly 2. This is analogous to the Wiegold Conjecture for finite simple groups.

The authors establish a logical chain of implications:
$$\text{Wiegold Conjecture (Finite)} \implies \text{Gelander's Conjecture (Algebraic)} \implies \text{Gelander's Conjecture (Compact Lie)}$$

3. Methodology

The paper's primary methodological innovation is the reduction of problems in continuous groups (Lie and algebraic) to problems in finite simple groups of Lie type via strong approximation and arithmetic reduction.

A. Arithmetic Reduction (The "One for Almost All" Trick)

The authors utilize the structure of algebraic groups defined over $\mathbb{Z}$.

1. Split $\mathbb{Z}$-schemes: Any connected simple complex algebraic group $G$ admits a structure as a split $\mathbb{Z}$-scheme.
2. Reduction Modulo $p$: For a prime $p$, one can reduce $G$ modulo $p$ to obtain a finite group $G_p(\mathbb{F}_p)$.
3. Strong Approximation: They prove that if a set $X$ generates the algebraic group $G$ (Zariski dense), then for almost all primes $p$, the reduction $X_p$ generates the finite group $G_p(\mathbb{F}_p)$.
4. Preservation of Irredundancy: Crucially, they show that if $X$ is irredundant (or Nielsen-irredundant) in $G$, then $X_p$ is irredundant in $G_p(\mathbb{F}_p)$ for almost all $p$.
5. Inequality: This leads to the fundamental inequality:
   $$m(G) \le \limsup_{p \to \infty} m(G_p(\mathbb{F}_p))$$
   This allows the authors to bound the continuous case using known (or estimated) bounds for finite groups.

B. Specialization and Isogenies

• Specialization: For tuples not initially in an arithmetic subgroup, they use a specialization lemma (Lemma 2.4) to map generators into a number field while preserving the irredundant generating property.
• Isogenies: They prove that the redundancy rank is invariant under isogenies (finite central quotients). Thus, $m(G) = m(\tilde{G}) = m(\bar{G})$ where $\tilde{G}$ is the simply-connected cover and $\bar{G}$ is the adjoint form.

C. Decomposition of Reductive and Amenable Groups

• Reductive Groups: They decompose reductive groups into a torus part $(\mathbb{G}_m)^r$ and a semisimple part. They prove additivity: $m(G_1 \times G_2) = m(G_1) + m(G_2)$ for simple factors.
• Amenable Lie Groups: Using the machinery of Abels and Noskov, they reduce the problem for general connected Lie groups to the study of Abels-Noskov groups (semidirect products of semisimple, abelian, and vector group parts). They show that $m(G)$ is finite if and only if $m(G/R_a(G))$ is finite, where $R_a(G)$ is the amenable radical.

4. Key Results and Theorems

Theorem 1.1 (Amenable Lie Groups)

For every connected amenable Lie group $G$, $m(G)$ is finite and bounded by a polynomial in the dimension:
$$m(G) \le a \cdot (\dim G)^b$$
Furthermore, $m(G) < \infty$ if and only if $m(G/R_a(G)) < \infty$.

Theorem 1.3 (Reductive Algebraic Groups)

For every connected reductive $\mathbb{C}$-algebraic group $G$, $m_{\mathbb{C}}(G)$ is finite and bounded by a polynomial in the rank:
$$m_{\mathbb{C}}(G) \le a \cdot (\text{rank}(G))^b$$

Theorem 1.5 (Connection to Finite Groups)

For a simple connected complex algebraic group $G$ and its compact real form $G_c(\mathbb{R})$:
$$m(G) \le m_{\mathbb{C}}(G) \le \limsup_{p} m(G_p(\mathbb{F}_p))$$
$$\mu(G) \le \mu_{\mathbb{C}}(G) \le \limsup_{p} \mu(G_p(\mathbb{F}_p))$$
This confirms that the bounds for continuous groups are controlled by the bounds for finite simple groups of Lie type.

Corollary 1.6 (Explicit Bounds via Harper's Theorem)

Using recent bounds by Harper ($m(G_p(\mathbb{F}_p)) \le 10^5 \cdot \text{rank}(G)^{10}$), the authors derive:

• Gelander's Conjecture holds for large $n$: If the number of generators $n > 10^5 \cdot \text{rank}(G)^{10}$, any generating tuple is redundant.
• Specific Values:
  • $\mu(SO(3)) = \mu(SL_2) = 2$ (Confirming Gelander's conjecture for these groups).
  • $m(SO(3)) = m(SL_2) = 3$.
  • $m(U(2)) = 4$.
  • $m(SO(4)) = 6$.
  • $m(SU(3)) \le 6$.

5. Significance and Contributions

1. Bridging Finite and Infinite Groups: The paper establishes a rigorous quantitative link between the generating properties of infinite Lie/algebraic groups and finite simple groups. This reduces the problem of bounding $m(G)$ in continuous settings to the active field of finite group theory.
2. Resolution of Conjectures: It provides strong evidence for Gelander's conjecture, proving it for specific low-rank groups and for all groups when the generating set size exceeds a polynomial bound in the rank.
3. Finiteness Criteria: It clarifies the structural conditions for the finiteness of $m(G)$. Specifically, $m(G)$ is finite if and only if the group is "amenable enough" (specifically, if the quotient by the amenable radical is finite in terms of redundancy, which essentially means the semisimple part is compact).
4. Methodological Distinction: The authors contrast their approach with independent work by Cantat, Dupont, and Martin-Baillon. While the latter uses Jordan's theorem and mirrors finite group arguments directly, Cohen and Vigdorovich use strong approximation to reduce to finite groups of Lie type, offering a different perspective on the same phenomenon.
5. Quantitative Bounds: The paper moves beyond qualitative "finiteness" to provide explicit polynomial bounds in terms of rank and dimension, which is a significant step forward in the quantitative theory of group generation.

In summary, the paper successfully characterizes the maximal size of irredundant generating sets for a broad class of Lie and algebraic groups, proving they are finite and polynomially bounded, and demonstrating that these bounds are intrinsically linked to the properties of finite simple groups.